Problem: A circle with radius $4$ has a sector with a $\dfrac{1}{4}\pi$ radian central angle. What is the area of the sector? ${16\pi}$ $\color{#9D38BD}{\dfrac{1}{4}\pi}$ ${2\pi}$ ${4}$
Answer: First, calculate the area of the whole circle. Then the area of the sector is some fraction of the whole circle's area. $A_c = \pi r^2$ $A_c = \pi (4)^2$ $A_c = 16\pi$ The ratio between the sector's central angle $\theta$ and $2 \pi$ radians is equal to the ratio between the sector's area, $A_s$ , and the whole circle's area, $A_c$ $\dfrac{\theta}{2 \pi} = \dfrac{A_s}{A_c}$ $\dfrac{1}{4}\pi \div 2 \pi = \dfrac{A_s}{16\pi}$ $\dfrac{1}{8} = \dfrac{A_s}{16\pi}$ $\dfrac{1}{8} \times 16\pi = A_s$ $2\pi = A_s$